* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
sel(0(),cons(X,Y)) -> X
sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
- Signature:
{activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,from,s,sel} and constructors {0,cons,n__from
,n__s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
sel(0(),cons(X,Y)) -> X
sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
- Signature:
{activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,from,s,sel} and constructors {0,cons,n__from
,n__s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
activate(x){x -> n__from(x)} =
activate(n__from(x)) ->^+ from(activate(x))
= C[activate(x) = activate(x){}]
** Step 1.b:1: InnermostRuleRemoval WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
sel(0(),cons(X,Y)) -> X
sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
- Signature:
{activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,from,s,sel} and constructors {0,cons,n__from
,n__s}
+ Applied Processor:
InnermostRuleRemoval
+ Details:
Arguments of following rules are not normal-forms.
sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
All above mentioned rules can be savely removed.
** Step 1.b:2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
sel(0(),cons(X,Y)) -> X
- Signature:
{activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,from,s,sel} and constructors {0,cons,n__from
,n__s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [2] x1 + [0]
p(cons) = [1] x1 + [1]
p(from) = [1] x1 + [0]
p(n__from) = [1] x1 + [0]
p(n__s) = [1] x1 + [1]
p(s) = [1] x1 + [0]
p(sel) = [1] x2 + [0]
Following rules are strictly oriented:
activate(n__s(X)) = [2] X + [2]
> [2] X + [0]
= s(activate(X))
sel(0(),cons(X,Y)) = [1] X + [1]
> [1] X + [0]
= X
Following rules are (at-least) weakly oriented:
activate(X) = [2] X + [0]
>= [1] X + [0]
= X
activate(n__from(X)) = [2] X + [0]
>= [2] X + [0]
= from(activate(X))
from(X) = [1] X + [0]
>= [1] X + [1]
= cons(X,n__from(n__s(X)))
from(X) = [1] X + [0]
>= [1] X + [0]
= n__from(X)
s(X) = [1] X + [0]
>= [1] X + [1]
= n__s(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
- Weak TRS:
activate(n__s(X)) -> s(activate(X))
sel(0(),cons(X,Y)) -> X
- Signature:
{activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,from,s,sel} and constructors {0,cons,n__from
,n__s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(activate) = [1] x1 + [0]
p(cons) = [1] x1 + [2]
p(from) = [1] x1 + [2]
p(n__from) = [1] x1 + [0]
p(n__s) = [1] x1 + [4]
p(s) = [1] x1 + [2]
p(sel) = [1] x1 + [9] x2 + [7]
Following rules are strictly oriented:
from(X) = [1] X + [2]
> [1] X + [0]
= n__from(X)
Following rules are (at-least) weakly oriented:
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__from(X)) = [1] X + [0]
>= [1] X + [2]
= from(activate(X))
activate(n__s(X)) = [1] X + [4]
>= [1] X + [2]
= s(activate(X))
from(X) = [1] X + [2]
>= [1] X + [2]
= cons(X,n__from(n__s(X)))
s(X) = [1] X + [2]
>= [1] X + [4]
= n__s(X)
sel(0(),cons(X,Y)) = [9] X + [26]
>= [1] X + [0]
= X
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
from(X) -> cons(X,n__from(n__s(X)))
s(X) -> n__s(X)
- Weak TRS:
activate(n__s(X)) -> s(activate(X))
from(X) -> n__from(X)
sel(0(),cons(X,Y)) -> X
- Signature:
{activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,from,s,sel} and constructors {0,cons,n__from
,n__s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [4]
p(activate) = [8] x1 + [2]
p(cons) = [1] x1 + [4]
p(from) = [1] x1 + [5]
p(n__from) = [1] x1 + [0]
p(n__s) = [1] x1 + [2]
p(s) = [1] x1 + [7]
p(sel) = [1] x1 + [1] x2 + [8]
Following rules are strictly oriented:
activate(X) = [8] X + [2]
> [1] X + [0]
= X
from(X) = [1] X + [5]
> [1] X + [4]
= cons(X,n__from(n__s(X)))
s(X) = [1] X + [7]
> [1] X + [2]
= n__s(X)
Following rules are (at-least) weakly oriented:
activate(n__from(X)) = [8] X + [2]
>= [8] X + [7]
= from(activate(X))
activate(n__s(X)) = [8] X + [18]
>= [8] X + [9]
= s(activate(X))
from(X) = [1] X + [5]
>= [1] X + [0]
= n__from(X)
sel(0(),cons(X,Y)) = [1] X + [16]
>= [1] X + [0]
= X
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:5: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(n__from(X)) -> from(activate(X))
- Weak TRS:
activate(X) -> X
activate(n__s(X)) -> s(activate(X))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
sel(0(),cons(X,Y)) -> X
- Signature:
{activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,from,s,sel} and constructors {0,cons,n__from
,n__s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(from) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [7] x1 + [0]
p(cons) = [1] x1 + [2]
p(from) = [1] x1 + [2]
p(n__from) = [1] x1 + [1]
p(n__s) = [1] x1 + [0]
p(s) = [1] x1 + [0]
p(sel) = [1] x2 + [4]
Following rules are strictly oriented:
activate(n__from(X)) = [7] X + [7]
> [7] X + [2]
= from(activate(X))
Following rules are (at-least) weakly oriented:
activate(X) = [7] X + [0]
>= [1] X + [0]
= X
activate(n__s(X)) = [7] X + [0]
>= [7] X + [0]
= s(activate(X))
from(X) = [1] X + [2]
>= [1] X + [2]
= cons(X,n__from(n__s(X)))
from(X) = [1] X + [2]
>= [1] X + [1]
= n__from(X)
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
sel(0(),cons(X,Y)) = [1] X + [6]
>= [1] X + [0]
= X
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:6: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
activate(X) -> X
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
s(X) -> n__s(X)
sel(0(),cons(X,Y)) -> X
- Signature:
{activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,from,s,sel} and constructors {0,cons,n__from
,n__s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^1))